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Cayley diagram

C2×(C19⋊C3)⋊C4 #90690

order456
isoclinismisoclinic class of D19⋊C3
automorphism?
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2

Aliases

direct productC2×(C19⋊C3)⋊C4
unambiguous semidirect product(C2×C19⋊C3)⋊C4
unambiguous semidirect product(C2×Dic19)⋊C3

Splittings

derived, SylowC19 ⋊ C2×C3×C4
HallC2×Dic19 ⋊ C3
HallC19⋊C3 ⋊ C2×C4

Subgroups

subgroupG/H
center, hypercenterC22D19⋊C3
commutator, hypocenterC19C2×C3×C4
FrattiniC2C2×D19⋊C3
Fitting, socleC22×C19C2×C3
prime radicalC22×C19⋊C3C2
cosocleC2×C19C22×C3
LCSG ▷ C19
UCSC1 ◁ C22

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 25
transitive degreeμt(G) = 152

A minimal faithful permutation representation — 2 generators on 25 points:

(3,4,5,6)(8,12,13,9,10,14)(11,20,17,15,16,22)(18,23,25,24,21,19);(1,2)(7,8,11,17,23,20,24,10,14,19,25,13,12,18,16,21,22,15,9)

Conjugacy classes & character table

36 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC4×(C19⋊C3)⋊C4, ((C19⋊C3)⋊C4)⋊C4, C4⋊((C19⋊C3)⋊C4), …
maximal normal subgroup ofC4×(C19⋊C3)⋊C4, ((C19⋊C3)⋊C4)⋊C4, C4⋊((C19⋊C3)⋊C4), …
maximal quotient ofC2×(C19⋊C3)⋊C8, #64520⋊C3, C4×(C19⋊C3)⋊C4, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(456, 10);